By Contradiction

Beyond the direct, contrapositive and "and/or" checks of the previous section, there are two further techniques for validating (or invalidating) a statement.

Method of contradiction. To show a statement p is true, begin by assuming the opposite — assume $\sim p$ is true — and reason forward from that assumption until it leads to something that clearly cannot be true (a contradiction of a known fact, or of the assumption itself). Since assuming $\sim p$ leads to an impossibility, $\sim p$ cannot be true, so p must be true after all.

Counter-example. To show a statement is false, it is enough to produce a single situation in which the statement fails to hold. Such a situation is called a counter-example — an example that counters, or defeats, the claim being made. One clean counter-example is all that is needed to disprove a statement, however plausible it may otherwise look. …